Five-loop renormalization group functions of ${O}(n) $-symmetric $\phi ...

Five-loop renormalization group functions of O(n)-symmetric φ4-theory and ǫ-expansions of arXiv:hep-th/9503230v1 1 Apr 1995

critical exponents up to ǫ5 H. Kleinert, J. Neu and V. Schulte-Frohlinde∗ Institut f¨ ur Theoretische Physik Freie Universit¨ at Berlin Arnimallee 14 D - 1000 Berlin 33

K.G. Chetyrkin and S.A. Larin Institute for Nuclear Research of the Academy of Sciences of the USSR 60th October Anniversary prospect 7a Moscow 117312, USSR

Abstract Motivated by the discovery of errors in six of the 135 diagrams in the published five-loop expansions of the β-function and the anomalous dimensions of the O(n)-symmetric φ4 -theory in D = 4 − ǫ dimensions we present the results of a full analytic reevaluation of all diagrams. The divergences are removed by minimal subtraction and ǫ-expansions are given for the critical exponents η, ν, and ω up to order ǫ5 .

∗

Work supported in part by Deutsche Forschungsgemeinschaft under grant no. Kl. 256.

1

1) During the last two decades, much effort has been invested into studying the scalar quantum field theory with φ4 -interaction. On the one hand, such a theory describes correctly many experimentally observable features of critical phenomena. Field theoretic renormalization group techniques [1] in D = 4 − ǫ dimensions [2, 3, 4] combined with Borel resummation methods of the resulting ǫ-expansions [5] led to extremely accurate determinations of the critical exponents of all O(n) universality classes. The latter requires the knowledge of the asymptotic behaviour of perturbation series in four dimensions which is completely known in this theory [6]. Apart from such important applications, the φ4 -theory, being the simplest renormalizable quantum field theory in the four dimensional space-time, has been an ideal ground for testing new methods of calculating Feynman diagrams and for studying the structure of perturbation theory. The RG functions of the φ4 -theory were first calculated analytically in four dimensions using dimensional regularization [7] and the minimal subtraction (MS) scheme [8] in the three- and four-loop approximations in Refs. [9] and [10]. The critical exponents were obtained as ǫ-expansions [3] up to terms of order ǫ3 and ǫ4 . The five-loop anomalous dimension of the field φ and the associated critical exponent η to order ǫ5 were determined analytically in [11]. The five-loop β-function and the anomalous dimension of the mass were given in Ref. [12]. However, three of the 124 four-point diagrams contributing to the β-function at the five-loop level could be evaluated only numerically. The analytic calculation of the β-function was finally completed in [13]. The ensuing ǫ-expansions for the critical exponents were obtained up to order ǫ5 in [14]. Intending further applications, the Berlin group of the authors undertook an independent recalculation of the perturbation series of Refs. [11, 12], using the same techniques, and discovered errors in six of the 135 diagrams. This meant that the subsequent results of [13, 14] were also incorrect. When visiting the Moscow group the errors were confirmed and we can now jointly report all expansions in the correct form. 2) We consider the O(n)-symmetric theory of n real scalar fields φa (a = 1, 2, . . . , n) with the Lagrangian 1 m2 16π 2 L = ∂µ φa ∂µ φa + B φa φa + gB (φa φa )2 2 2 4!

(1)

in an euclidean space with D = 4−ǫ dimensions. The bare (unrenormalized) coupling 2

constant gB and mass mB are expressed via renormalized ones as gB = µǫ Zg g = µǫ

Z4 g, (Z2 )2

m2B = Zm2 m2 =

Z φ2 2 m . Z2

(2)

Here µ is the unit of mass in dimensional regularization and Z4 , Z2 , Zm2 are the renormalization constants of the vertex function, propagator and mass, respectively, with Zφ2 being the renormalization constant of the two-point function obtained from the propagator by the insertion, in all possible ways, of the vertex (−φ2 ) [10]. In the MS-scheme the renormalization constants do not depend on dimensional parameters and are expressible as series in 1/ǫ with purely g-dependent coefficients: Zi = 1 +

∞ X

Zi,k (g) . ǫk k=1

(3)

The β-function and the anomalous dimensions entering the RG equations are expressed in the standard way as follows:

1 ∂ Zg,1 d g ǫ = g , β(g) = g + 2 2 d ln µ gB 2 ∂ g

d ln m d ln Zm2 1 ∂ Zm2 ,1 γm = =− = g , 2 d ln µ gB d ln µ 2 ∂g

d ln Zi 1 ∂ Zi,1 γi (g) = = − g , d ln µ2 gB 2 ∂g

i = 2, 4, φ2 .

(4)

(5) (6)

We also use the relations

β(g) = g[2γ2(g) − γ4 (g)] ,

γm (g) = γ2 (g) − γφ2 (g),

(7)

which follow from the relations between renormalization constants implied by (2) and are useful for the calculations of β(g) and γm (g). To determine all RG functions up to five loops we calculate the five-loop approximation to the three constants Z2 , Z4 and Zφ2 . The constant Z2 contains the counterterms of the 11 five-loop propagator diagrams. Two of them were calculated erroneously in Ref. [11]. The constant Z4 receives contributions from 124 vertex diagrams. Of these diagrams, 90 contribute to Zφ2 after appropriate changes of combinatorial factors. Four of the 124 counterterms were calculated erroneously in Ref. [12]. In the present paper we have used the same methods as in the previous works [11, 12] to calculate the counterterms from the dimensionally regularized Feynman integrals, namely, the method of infrared rearrangement [15], the Gegenbauer 3

polynomial x-space technique (GPXT) [16], the integration-by-parts algorithm [17], and the R∗ -operation [18]. Three diagrams were calculated analytically first in [13] by using the so-called method of uniqueness, later the same results were obtained for them by using the Gegenbauer polynomials in x-space together with several nontrivial tricks [19]. A detailed description of the calculations including the diagramwise results will be presented in a separate publication. The analytic results of our recalculation of the five-loop approximations to the RG functions β(g), γ2 (g) and γm (g) are [ζ(n) is the Riemann ζ-function]: β(g) =

g2 6 [n +

g3 6 [3n

8] −

+ 14]

4

g + 432 [33n2 + 922n + 2960 + ζ(3) · 96(5n + 22)] 5

g [−5n3 + 6320n2 + 80456n + 196648 − 7776

+ ζ(3) · 96(63n2 + 764n + 2332) − ζ(4) · 288(5n + 22)(n + 8) + ζ(5) · 1920(2n2 + 55n + 186)] 6

g + 124416 [13n4 + 12578n3 + 808496n2 + 6646336n + 13177344

+ ζ(3) · 16(−9n4 + 1248n3 + 67640n2 + 552280n + 1314336) + ζ 2 (3) · 768(−6n3 − 59n2 + 446n + 3264) − ζ(4) · 288(63n3 + 1388n2 + 9532n + 21120) + ζ(5) · 256(305n3 + 7466n2 + 66986n + 165084) − ζ(6)(n + 8) · 9600(2n2 + 55n + 186) + ζ(7) · 112896(14n2 + 189n + 526)] ,

γ2 (g) =

g2 36 (n

+ 2) −

g3 432 (n

(8)

+ 2)[n + 8]

4

g + 5184 (n + 2)

[5(−n2 + 18n + 100)]

5

g (n + 2)[39n3 + 296n2 + 22752n + 77056 − 186624

− ζ(3) · 48(n3 − 6n2 + 64n + 184) + ζ(4) · 1152(5n + 22)] ,

γm (g) =

g 6 (n

+ 2) −

g2 36 (n

+ 2)[5] +

g3 72 (n

+ 2)[5n + 37]

4

g (n + 2) [−n2 + 7578n + 31060 − 15552

+ ζ(3) · 48(3n2 + 10n + 68)

4

(9)

+ ζ(4) · 288(5n + 22)] g5

+ 373248 (n + 2)[21n3 + 45254n2 + 1077120n + 3166528 + ζ(3) · 48(17n3 + 940n2 + 8208n + 31848) − ζ 2 (3) · 768(2n2 + 145n + 582) + ζ(4) · 288(−3n3 + 29n2 + 816n + 2668) + ζ(5) · 768(−5n2 + 14n + 72) + ζ(6) · 9600(2n2 + 55n + 186)] .

(10)

For n = 1 the series have the numerical form: β(g) = 1.5 g 2 − 2.833 g 3 + 16.27 g 4 − 135.8 g 5 + 1424.2841 g 6 ,

(11)

γ2 = 0.0833 g 2 − 0.0625 g 3 + 0.3385 g 4 − 1.9256 g 5 ,

(12)

γm = 0.5 g − 0.4167 g 2 + 1.75 g 3 − 9.978 g 4 + 75.3778 g 5 .

(13)

Note that the five-loop coefficients have changed by about 0.3 % for the β-function, by about 9 % for γm , and by a factor of three for γ2 in comparison with the wrong results of Refs. [11, 12]. 3) These RG functions can now be used to calculate the critical exponents describing the behaviour of a statistical system near the critical point of the second order phase transition [4]. At the critical temperature T = TC , the asymptotic behaviour of the correlation function for |x| → ∞ has the form Γ(x) ∼

1 D−2+η

|x|

.

(14)

Close to TC , the correlation length behaves for t = T − TC → 0 as ξ ∼ t−ν (1 + const · tων + . . .) .

(15)

The three critical exponents η, ν and ω defined in this way completely specify the critical behaviour of the system. All other exponents can be expressed in terms of these [4]. The three critical exponents can be determined from the RG functions of the 4

φ -theory by going to the infrared-stable fixed point g = g0 (ǫ) =

∞ X

k=1

5

g (k) ǫk

(16)

which is determined by the condition (βǫ ≡ β − 2ǫ g) βǫ′ (g0 ) = 0,

βǫ (g0 ) = [∂βǫ (g)/∂g]g=g0 > 0 .

(17)

The resulting formulas for the critical exponents are: η = 2γ2(g0 ) ,

1/ν = 2(1 − γm (g0 )) ,

w = 2βǫ′ (g0 ) ,

(18)

each emerging as an ǫ-expansion up to order ǫ5 . From (8)-(10) we therefore find: η(ǫ) =

(n+2)ǫ2 2(n+8)2

n

ǫ 2 4(n+8)2 [−n

1+

+ 56n + 272]

2

ǫ 4 3 2 − 16(n+8) 4 [5n + 230n − 1124n − 17920n − 46144

+ ζ(3)(n + 8) · 384(5n + 22)] 3

ǫ 6 5 4 3 − 64(n+8) 6 [13n + 946n + 27620n + 121472n

−262528n2 − 2912768n − 5655552 − ζ(3)(n + 8) · 16(n5 + 10n4 + 1220n3 − 1136n2 −68672n − 171264) + ζ(4)(n + 8)3 · 1152(5n + 22) o

− ζ(5)(n + 8)2 · 5120(2n2 + 55n + 186)] ,

1/ν(ǫ) = 2 +

(n+2)ǫ n+8

n

−1 −

ǫ 2(n+8)2 (13n

+ 44)

ǫ2

+ 8(n+8)4 [3n3 − 452n2 − 2672n − 5312 + ζ(3)(n + 8) · 96(5n + 22)] ǫ3

+ 32(n+8)6 [3n5 + 398n4 − 12900n3 − 81552n2 − 219968n − 357120 + ζ(3)(n + 8) · 16(3n4 − 194n3 + 148n2 + 9472n + 19488) + ζ(4)(n + 8)3 · 288(5n + 22) − ζ(5)(n + 8)2 · 1280(2n2 + 55n + 186)] 4

ǫ 7 6 5 4 + 128(n+8) 8 [3n − 1198n − 27484n − 1055344n

−5242112n3 − 5256704n2 + 6999040n − 626688 − ζ(3)(n + 8) · 16(13n6 − 310n5 + 19004n4 + 102400n3 −381536n2 − 2792576n − 4240640) − ζ 2 (3)(n + 8)2 · 1024(2n4 + 18n3 + 981n2 + 6994n + 11688) + ζ(4)(n + 8)3 · 48(3n4 − 194n3 + 148n2 + 9472n + 19488) + ζ(5)(n + 8)2 · 256(155n4 + 3026n3 + 989n2 − 66018n − 130608)

6

(19)

− ζ(6)(n + 8)4 · 6400(2n2 + 55n + 186) o

+ ζ(7)(n + 8)3 · 56448(14n2 + 189n + 526)] ,

ω(ǫ) = ǫ −

ǫ2 (n+8)2 [9n

(20)

+ 42]

3

ǫ 3 2 + 4(n+8) 4 [33n + 538n + 4288n + 9568

+ ζ(3)(n + 8) · 96(5n + 22)] 4

ǫ 5 4 3 2 + 16(n+8) 6 [5n − 1488n − 46616n − 419528n − 1750080n − 2599552

− ζ(3)(n + 8) · 96(63n3 + 548n2 + 1916n + 3872) + ζ(4)(n + 8)3 · 288(5n + 22) − ζ(5)(n + 8)2 · 1920(2n2 + 55n + 186)] 5

ǫ 7 6 5 4 + 64(n+8) 8 [13n + 7196n + 240328n + 3760776n

+38877056n3 + 223778048n2 + 660389888n + 752420864 − ζ(3)(n + 8) · 16(9n6 − 1104n5 − 11648n4 − 243864n3 −2413248n2 − 9603328n − 14734080) − ζ 2 (3)(n + 8)2 · 768(6n4 + 107n3 + 1826n2 + 9008n + 8736) − ζ(4)(n + 8)3 · 288(63n3 + 548n2 + 1916n + 3872) + ζ(5)(n + 8)2 · 256(305n4 + 7386n3 + 45654n2 + 143212n + 226992) − ζ(6)(n + 8)4 · 9600(2n2 + 55n + 186) + ζ(7)(n + 8)3 · 112896(14n2 + 189n + 526)] .

(21)

For n = 1, these expansions read, numerically, η = 0.01852 ǫ2 + 0.01869 ǫ3 − 0.00833 ǫ4 + 0.02566 ǫ5 , 1 ν

= 2 − 0.333 ǫ − 0.1173 ǫ2 + 0.1245 ǫ3 − 0.307 ǫ4 + 0.951 ǫ5 , ω = ǫ − 0.630 ǫ2 + 1.618 ǫ3 − 5.24 ǫ4 + 20.75 ǫ5 .

(22) (23) (24)

Note that η (5) has decreased by about 30 % in comparison with the (incorrect) results of Ref. [14], ν (5) has increased by about 10 %, and ω (5) increased by about 0.6 % in comparison with [14]. It is known that the series of the ǫ-expansion are asymptotic series and special resummation techniques [20, 21] should be applied to obtain reliable estimates of the critical exponents. Although the size of the ǫ5 terms in the physical dimension 7

(i.e., at ǫ = 1) is very large, their contribution to the exponents in the resummed series is very small. This is why even large relative changes of the ǫ5 coefficients turn out not to change the critical exponents [22] within the accuracy of previous determinations [14, 23].

References [1] N.N.Bogoliubov and D.V.Shirkov, Introduction to the Theory of Quantized Fields, Interscience, New York, 1980. [2] K.G.Wilson, Phys.Rev. B 4 (1971) 3184; K.G.Wilson and J.B.Kogut, Phys.Rep. C 12 (1974) 75. [3] K.G.Wilson and M.E.Fisher, Phys.Rev.Lett. 28 (1972) 240. [4] E.Brezin, J.C.Le Guillou and J.Zinn-Justin, in Phase Transitions and Critical Phenomena ed. by C.Domb and M.S.Green, Academic Press, New York, 1976, vol.6 [5] J.Zinn-Justin, Quantum Field Theory and Critical Phenomena, Clarendon Press, Oxford, 1989 [6] L.N.Lipatov, JETP 72 (1977) 411. [7] G.’t Hooft and M.Veltman, Nucl.Phys. B 44 (1972) 189. [8] G.’t Hooft, Nucl.Phys. B 61 (1973) 455. [9] E.Brezin, J.C.Le Guillou, J.Zinn-Justin and B.G.Nickel, Phys.Lett. A 44 (1973) 227. [10] A.A.Vladimirov, D.I.Kazakov and O.V.Tarasov, Sov. Phys. JETP 50 (1979) 521; Preprint JINR E2-12249, Dubna, 1979. [11] K.G.Chetyrkin, A.L.Kataev and F.V.Tkachov, Phys.Lett. B 99 (1981) 147; ibid. 101 (1981) 457 (Erratum). [12] K.G.Chetyrkin, S.G.Gorishny, S.A.Larin and F.V.Tkachov, Phys.Lett. B 132 (1983) 351; Preprint INR P-0453, Moscow, 1986. 8

[13] D.I.Kazakov, Phys.Lett. B 133 (1983) 406; Teor.Mat.Fiz. 58 (1984) 343; Dubna lecture notes, E2-84-410 (1984). [14] S.G.Gorishny, S.A.Larin and F.V.Tkachov, Phys.Lett. A 101 (1984) 120. [15] A.A.Vladimirov, Teor.Mat.Fiz. 43 (1980) 210. [16] K.G.Chetyrkin and F.V.Tkachov, Preprint INR P-118, Moscow, 1979; K.G.Chetyrkin, A.L.Kataev and F.V.Tkachov, Nucl.Phys. B 174 (1980) 345. [17] F.V.Tkachov, Phys.Lett. B 100 (1981) 65; K.G.Chetyrkin and F.V.Tkachov, Nucl.Phys. B 192 (1981) 159. [18] K.G.Chetyrkin and F.V.Tkachov, Phys.Lett. B 114 (1982) 340; K.G.Chetyrkin and V.A. Smirnov, Phys.Lett. B 144 (1984) 419. [19] D.J. Broadhurst, Massless Scalar Feynman Diagrams: Five Loops and Beyond , Open University Preprint OUT - 4102 - 18 1985, Milton Keynes, U.K. [20] D.I.Kazakov and D.V.Shirkov, Fortschr.Phys. 28 (1980) 465; J.Zinn-Justin, Phys.Rep. 70 (1981) 3. [21] W. Janke and H. Kleinert, Resummation of Divergent Perturbation Series, World Scientific, Singapore, 1991. [22] This was checked in detail by W. Janke and H. Kleinert using the resummation programs of Ref. [22]. Since the deviations from the previous numbers were found to be very small the results were not published. [23] J.C.Le Guillou and J.Zinn-Justin, J.Physique Lett. 46 (1985) 137.

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